Question

You have just won the lottery and you can choose between the following payout options. The annual interest rate (EAR) is 10%. a) $100,000 right now and $60,000 every two years starting 3 years from now and ending 17 years from now (i.e., payments are at t = 0, t = 3, t = 5, … , t = 15, t = 17). b) $60,000 a year for 25 years with the first payment one year from today (i.e., payments are at t = 1, 2, … 24, 25). c) 25 annual payments of $45,000 and a 26th payment of $299,000. The first payment is made right now, and the $299,000 payment is made one year after the last $45,000 payment. How much more is the best option worth today relative to the worst option?

Select one:

a. $920,000

b. $276,678

c. $241,409

d. $221,088

e. $235,871

f. $191,898

g. $235,323

h. -$67,712

NOTE: Please show all the work, not using EXCEL. (Step by step with equations) Thanks!

Answer 1

To know the worth of any option "today", we have to calculate the Present Value (discounted value) of the all the cash flows in each of the options

Discount Rate: 10%

Note: CF_x means cash flow occurring at time, t=x

Discount Factor for time,DF_x at t=x = 1/((1+10%)^^x)

Option A:

CF₀ = 100000 $; CF₃ = 60000 $; CF₅ = 60000; CF₇ = 60000; CF₉ = 60000; CF₁₁ = 60000; CF₁₃ = 60000; CF₁₅= 60000; CF₁₇= 60000

Discount factor for each cash flow = 1/(1+10%)^⁰; 1/(1+10%)^³; 1/(1+10%)^⁵; 1/(1+10%)^⁷; 1/(1+10%)^⁹; 1/(1+10%)^¹¹; 1/(1+10%)^¹³; (1+10%)^^15; (1+10%)^¹⁷= 1 ; 0.76; 0.63; 0.52; 0.43; 0.36; 0.29; 0.24; 0.20

Present Value = Sum of CF_x * DF_x for x = 0 to t

Therefore, for Option A, Present Value = 100000*1+60000*0.76+60000*0.63+60000*0.52+60000*0.43+60000*0.36+60000*0.29+60000*0.24+60000*0.2
`=3,02,800 $

Similarly for option B

Present Value = 60000*0.91+60000*0.83+60000*0.75+60000*0.68+60000*0.62+60000*0.56+60000*0.51+60000*0.47+60000*0.42+60000*0.39+60000*0.35+60000*0.32+60000*0.29+60000*0.26+60000*0.24+60000*0.22+60000*0.2+60000*0.18+60000*0.16+60000*0.15+60000*0.14+60000*0.12+60000*0.11+60000*0.1+60000*0.09+*
= 5,44,200 $

For Option C

Present Value

= 45000*1+45000*0.91+45000*0.83+45000*0.75+45000*0.68+45000*0.62+45000*0.56+45000*0.51+45000*0.47+45000*0.42+45000*0.39+45000*0.35+45000*0.32+45000*0.29+45000*0.26+45000*0.24+45000*0.22+45000*0.2+45000*0.18+45000*0.16+45000*0.15+45000*0.14+45000*0.12+45000*0.11+45000*0.1+299000*0.09
= 4,76,010 $

Best Option is Option B Present Value = 5,44,200 $

Worst Option is Option A Present Value = 3,02,800 $

Difference in best option and worst option = 5,44,200 - 3,02,800 = 2,41,400 $

Therefore, correct option is Option C.

Note: The answer does not match to the last digit as the calculations are done bot using excel and by rounding to two digits.